On spaces which are linearly D

A note on transitively D-spaces

“…In [16], Guo and Junnila introduced the concept of linearly D. A family U of subsets of X is monotone if U is linearly ordered by inclusion. A neighborhood assignment ϕ for X is monotone provided that {ϕ(x) : x ∈ X} is a monotone family.…”

“…A space X is linearly D provided that for every monotone neighborhood assignment ϕ for X there exists a closed discrete subspace D of X such that X = {ϕ(d) : d ∈ D} (cf. [16]). By conclusions which appears in [15], we know that a space X is linearly Lindelöf if and only if X is linearly D and has countable extent.…”

“…[23]). The linear Dspaces are called transitively D-spaces in [21], since the concept of linearly D has a proper meaning in [16]. A transitive neighborhood assignment for a space X is a function ϕ from X to the topology of the space X such that x ∈ ϕ(x) and for each y ∈ ϕ(x) we have ϕ(y) ⊆ ϕ(x) for any x ∈ X.…”

Properties D and aD are different

“…Before that we consider another weakening of property D. Recently H. Guo and H.J.K. Junnila in [7] introduced the notion of linearly D-spaces and proved several nice results concerning the topic. In Sections 3 and 4 we answer the following two questions from [7] in negative (in ZFC): Problem 2.5.…”

“…Junnila in [7] introduced the notion of linearly D-spaces and proved several nice results concerning the topic. In Sections 3 and 4 we answer the following two questions from [7] in negative (in ZFC): Problem 2.5. Let X be a T 1 (linearly) D-space and let A ⊆ X have uncountable regular cardinality.…”